3 research outputs found
Towards Logical Specification of Statistical Machine Learning
We introduce a logical approach to formalizing statistical properties of
machine learning. Specifically, we propose a formal model for statistical
classification based on a Kripke model, and formalize various notions of
classification performance, robustness, and fairness of classifiers by using
epistemic logic. Then we show some relationships among properties of
classifiers and those between classification performance and robustness, which
suggests robustness-related properties that have not been formalized in the
literature as far as we know. To formalize fairness properties, we define a
notion of counterfactual knowledge and show techniques to formalize conditional
indistinguishability by using counterfactual epistemic operators. As far as we
know, this is the first work that uses logical formulas to express statistical
properties of machine learning, and that provides epistemic (resp.
counterfactually epistemic) views on robustness (resp. fairness) of
classifiers.Comment: SEFM'19 conference paper (full version with errors corrected
Statistical Epistemic Logic
We introduce a modal logic for describing statistical knowledge, which we
call statistical epistemic logic. We propose a Kripke model dealing with
probability distributions and stochastic assignments, and show a stochastic
semantics for the logic. To our knowledge, this is the first semantics for
modal logic that can express the statistical knowledge dependent on
non-deterministic inputs and the statistical significance of observed results.
By using statistical epistemic logic, we express a notion of statistical
secrecy with a confidence level. We also show that this logic is useful to
formalize statistical hypothesis testing and differential privacy in a simple
and abstract manner